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Properties

Label 20.0.38698352640...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 3^{10}\cdot 5^{22}$
Root discriminant $37.97$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79524, 0, 113640, 0, 70000, 0, 9360, 0, -10600, 0, -4724, 0, -360, 0, 160, 0, 60, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 + 60*x^16 + 160*x^14 - 360*x^12 - 4724*x^10 - 10600*x^8 + 9360*x^6 + 70000*x^4 + 113640*x^2 + 79524)
 
gp: K = bnfinit(x^20 + 10*x^18 + 60*x^16 + 160*x^14 - 360*x^12 - 4724*x^10 - 10600*x^8 + 9360*x^6 + 70000*x^4 + 113640*x^2 + 79524, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} + 60 x^{16} + 160 x^{14} - 360 x^{12} - 4724 x^{10} - 10600 x^{8} + 9360 x^{6} + 70000 x^{4} + 113640 x^{2} + 79524 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(38698352640000000000000000000000=2^{38}\cdot 3^{10}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.97$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{76} a^{16} + \frac{1}{38} a^{14} - \frac{9}{76} a^{12} + \frac{3}{19} a^{10} - \frac{9}{19} a^{8} + \frac{5}{19} a^{6} - \frac{9}{19} a^{4} - \frac{1}{2} a^{2} + \frac{3}{19}$, $\frac{1}{76} a^{17} + \frac{1}{38} a^{15} - \frac{9}{76} a^{13} + \frac{3}{19} a^{11} - \frac{9}{19} a^{9} + \frac{5}{19} a^{7} - \frac{9}{19} a^{5} - \frac{1}{2} a^{3} + \frac{3}{19} a$, $\frac{1}{4885088307369105588} a^{18} + \frac{12025673411236861}{4885088307369105588} a^{16} - \frac{102452584097992577}{1628362769123035196} a^{14} + \frac{192176297666304227}{2442544153684552794} a^{12} - \frac{26516872508211431}{814181384561517598} a^{10} + \frac{467039458965027421}{1221272076842276397} a^{8} - \frac{1117429033123065485}{2442544153684552794} a^{6} - \frac{157296144547995335}{814181384561517598} a^{4} - \frac{488537194441171160}{1221272076842276397} a^{2} - \frac{66128600766299157}{407090692280758799}$, $\frac{1}{688797451339043887908} a^{19} + \frac{469018131884730472}{172199362834760971977} a^{17} - \frac{11846143455236244649}{114799575223173981318} a^{15} + \frac{16968926052575128651}{172199362834760971977} a^{13} + \frac{18880302610164716}{57399787611586990659} a^{11} - \frac{3323714019864483451}{18126248719448523366} a^{9} - \frac{6504381206451562570}{172199362834760971977} a^{7} + \frac{232026403413949837}{57399787611586990659} a^{5} - \frac{28577794961813528291}{172199362834760971977} a^{3} + \frac{16046092483187943835}{57399787611586990659} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 38333981.767314285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}, \sqrt{-15})\), 5.1.50000.1, 10.2.5120000000000.4, 10.0.3037500000000.2, 10.0.6220800000000000.8

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed